Real hypersurfaces in complex hyperbolic two-plane Grassmannians related to the Reeb vector field

TL;DR

This paper characterizes certain real hypersurfaces in noncompact complex two-plane Grassmannians, showing they are either tubes over totally real submanifolds or horospheres with specific geometric properties.

Contribution

It provides a new characterization of real hypersurfaces with Reeb vector field in the maximal quaternionic subbundle, identifying them as tubes over totally real submanifolds or horospheres.

Findings

Hypersurfaces are tubes over totally real submanifolds.

Hypersurfaces can be horospheres with singular centers.

Classification depends on the Reeb vector field's position.

Abstract

In this paper we give a characterization of real hypersurfaces in noncompact complex two-plane Grassmannian $SU_{2,m}/S(U_2 U_m)$, $m \geq 2$ with Reeb vector field $\xi$ belonging to the maximal quaternionic subbundle $\mathcal Q$. Then it becomes a tube over a totally real totally geodesic ${\mathbb H}H^n$, $m=2n$, in noncompact complex two-plane Grassmannian $SU_{2,m}/S(U_2 U_m)$, a horosphere whose center at the infinity is singular or another exceptional case.